Baghdad was in decline and superseded by cities like Cairo long before the Mongols turned up. Some historians have traced this decline all the way to the ninth century CE, when Abbasids failed to bring land administration and irrigation in Iraq up to pre-Islamic standards. Then decentralisation, a weakened Caliph, and most importantly the Iqta’ system and the corruption it brought with it made it worse. Even though there were some serious efforts made by the Buyids to restore public buildings, by the tenth century many of the palatial mansions of the Abbasid golden age were practically slums. Based on the vast biographical records of scholars, we know that there was a general exodus to other safer and more prosperous regions. A medieval “brain drain”. Meanwhile, cities like Cairo or Damascus continues to thrive. Especially Cairo, which enjoyed its status as the main node of the Red Sea and Indian Ocean-Mediterranean trade. A lot of comparisons could be made between Baghdad in the eve of the Mongol invasion and Rome when it was finally sacked by the Vandals. Neither of these invading forces were as “barbaric” as imagined (both had been quite Islamised or Romanised) by then, and nether of them would have been walking into thriving cities. After the invasion, the centre of power split more evenly between the Western and Eastern halves of the Muslim world, and Baghdad was no longer the Caliphal capital (the Mamluks brought the Abbasids to Cairo - imagine the Avignon papacy becoming de facto papacy). Instead, it was simply a frontier town with barely a glimmer of its glorious past. Perhaps if the Mamluks or Ottomans had succeeded in conquering their Eastern rivals the Timirids or the Safavids, then maybe Baghdad could have been restored as the centre of the Muslim caliphate. However, theologically, there are no clear reasons to do this, unlike Rome. It is only when the British started fanning the flames of Arab nationalism in WWI that people started thinking about restoring Baghdad as the centre of a new Arab state.

EDIT: I forgot to include any references to the discussion. David Waines’ “The Third Century Internal Crisis of the Abbasids”, Journal of Economic and Social History of the Orient 20.3 (1977), pp.282-386 is still a good place to get a picture of how bad things were already early on. https://www.jstor.org/stable/3631960 The rest is speculation on my part, but I can provide references to general reading on the many periods covered here if requested. There is still no decent long duree study of Baghdad in the Middle Ages.